PHYSICAL REVIEW E 77, 061138 共2008兲
Intrinsic and extrinsic noise effects on phase transitions of network models with applications to swarming systems Jaime A. Pimentel,1 Maximino Aldana,1,* Cristián Huepe,2 and Hernán Larralde1
1
Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México, Apartado Postal 48-3, Cuernavaca, Morelos 62251, Mexico 2 614 North Paulina Street, Chicago, Illinois 60622-6062, USA 共Received 18 February 2008; published 27 June 2008兲 We analyze order-disorder phase transitions driven by noise that occur in two kinds of network models closely related to the self-propelled model proposed by Vicsek et al. 关Phys. Rev. Lett. 75, 1226 共1995兲兴 to describe the collective motion of groups of organisms. Two different types of noise, which we call intrinsic and extrinsic, are considered. The intrinsic noise, the one used by Vicsek et al. in their original work, is related to the decision mechanism through which the particles update their positions. In contrast, the extrinsic noise, later introduced by Grégoire and Chaté 关Phys. Rev. Lett. 92, 025702 共2004兲兴, affects the signal that the particles receive from the environment. The network models presented here can be considered as mean-field representations of the self-propelled model. We show analytically and numerically that, for these two network models, the phase transitions driven by the intrinsic noise are continuous, whereas the extrinsic noise produces discontinuous phase transitions. This is true even for the small-world topology, which induces strong spatial correlations between the network elements. We also analyze the case where both types of noise are present simultaneously. In this situation, the phase transition can be continuous or discontinuous depending upon the amplitude of each type of noise. DOI: 10.1103/PhysRevE.77.061138
PACS number共s兲: 64.60.⫺i, 05.70.Fh, 05.70.Ln, 87.17.Jj
I. INTRODUCTION
In spite of many efforts, only a limited understanding has been achieved regarding the emergence of collective order in nonequilibrium systems. While these systems often present features analogous to the those found in equilibrium, such as phase transitions and long-range correlations, it is not clear how to use the powerful tools available in equilibrium statistical mechanics to analyze their properties. To overcome this problem, it may be useful to consider cases where simple qualitative characteristics 共e.g., the existence and order of a phase transition兲 are common to different nonequilibrium systems with similar properties. A class of nonequilibrium systems that has sparked increasing interest in recent years is given by models of groups of swarming agents 关1–8兴. These are used to describe the collective behavior of self-propelled agents such as schools of fish, flocks of birds, or herds of quadrupeds 关9–11兴. Even the simplest of these models displays large-scale organized structures, in which agents separated by distances much larger than their interaction ranges can coordinate and swarm in the same direction. If noise is added to the system, this ordered state is destroyed as the noise level increases. When the noise reaches a critical value, the system undergoes a phase transition to a disordered state where agents move in random directions 关2,3兴. This phase transition has been quite thoroughly analyzed through numerical simulations. However, the lack of a systematic theoretical approach to nonequilibrium systems has hindered a proper characterization of the order-disorder phase transition, and there are still doubts about its basic features even in this simplest of cases 关12–14兴.
*
[email protected] 1539-3755/2008/77共6兲/061138共17兲
In this paper we are particularly interested in how this phase transition might be affected by the way in which the noise is introduced in the system. Two different types of noise, which we will call extrinsic and intrinsic, have recently been considered in models of swarming 关2,12兴. In these models, at every time step each particle receives a signal from its neighbors that tells the particle in which direction to move next. The extrinsic noise consists in that the signal received by the particle is blurred 共because, say, the environment is not completely transparent and the particle cannot see its neighbors very well兲. As a consequence, the particle may move in a different direction to the one dictated by the neighbors. In contrast, in the intrinsic noise case each particle receives the signal sent by the neighbors perfectly, but then it may “decide” to do something else and move in a different direction. Thus, the extrinsic noise can be thought of as produced by a blurry environment, whereas the intrinsic noise comes from the “free will” of the particles, so to speak; namely, from the uncertainty in the particle’s decision mechanism. In either case, of course, the net result is that, at every time step, the particle may move in a direction that departs from the one dictated by the neighbors. It has been pointed out that these two distinct types of noise, extrinsic and intrinsic, can produce very different order-disorder phase transitions 关14兴. This can been shown analytically using a network approach in which the elements, instead of interacting with the neighbors in a physical space, interact with any element that is linked to them through a network connection. This kind of description has been used to model a large range of dynamics, such as the traffic between Internet websites or servers, the evolution of an epidemic outbreak, the mechanisms triggered by gene expressions in the cell, or the activity of the brain 关15–20兴. In the context of swarming systems, the network approach is equivalent to a mean-field theory in which correlations be-
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tween the particles are not taken into account. However, this approach has the virtue that it allows us to separate clearly the dynamical interaction rule that determines the dynamical state of the particles, from the topology of the underlying network that develops in time and space and dictates who interacts with who. Therefore, under the network approach it is possible to focus on the effects that the two different types of noise have on the dynamics of the system. Even when the network approach leaves aside some important aspects of the dynamics of swarming systems 共such as correlations in space and time兲, some appealing analogies can be established between the swarming and network systems. Indeed, in the simplest swarming models, the dynamics is defined by giving to each agent a steering rule that uses the velocities of all agents in its vicinity as an input to compute its own velocity for the next time step. This algorithm can be associated to a dynamics on a switching network that links at every time step all agents that are within the interaction range of each other. In this context, the network is simply a representation of the spatial dynamics of the system. However, it has been shown that this analogy can be pushed further successfully and that a static network with long-range connections can capture some of the main qualitative behaviors of simple swarming models 关14,21,22兴. In this paper, we compare the properties of the phase transitions and dynamical mappings of two kinds of network models to further explore the analogies described above. We consider models that incorporate three of the main aspects of the interaction between the particles in swarms: an average input signal from the neighbors, noise, and, in some sense, extremely long-range interactions. In the first kind of model the elements of the network can acquire only two states +1 and −1; whereas in the second, the elements are represented by 2D vectors whose angles take any value between 0 and 2. We find that swarming systems and their network counterparts indeed present qualitatively similar behaviors depending on whether the noise is intrinsic or extrinsic. We also determine numerically that the same qualitative features arise when the particles are placed on a small-world network, and we extend our results to the case in which the network models are subject to both types of noise. The paper is organized as follows. In Sec. II we present the model introduced by Vicsek and his group to describe the emergence of order in swarming systems. In particular, we focus our attention on how the phase transition seems to change when the noise changes from intrinsic to extrinsic. In Sec. III we present a majority voter model on a network, which is reminiscent of the Ising model with discrete internal degrees of freedom. This model is simple enough as to be treated analytically, at least for the case of homogeneous random network topologies for which we show analytically that the two types of noise indeed produce two different types of phase transition. In Sec. IV we introduce another network model in which the internal degrees of freedom are continuous 共2D vectors兲. This model can be treated analytically in the limit of infinite network connectivity. However, these results and extensive numerical simulations clearly indicate that the two types of noise again produce two different phase transitions, which are analogous to the ones observed in the majority voter model. In Sec. V we discuss the mean-
field assumptions conveyed in the two network models and how they relate to the self-propelled model. We also show that the nature of the phase transition produced by each type of noise does not change when the small-world topology is implemented, which includes strong spatial correlations between the network elements. Finally, in Sec. VI we summarize our results. II. THE VICSEK MODEL
Arguably, the simplest model to describe the collective motion of a group of organisms was proposed by Vicsek and his collaborators 关2兴. In this model, N particles move within a 2D box of sides L with periodic boundary conditions. The particles are characterized by their positions xជ 1共t兲 , . . . , xជ N共t兲 and their velocities vជ 1共t兲 = vei1共t兲 , . . . , vជ N共t兲 = veiN共t兲 共represented here as complex numbers兲. All the particles move with the same speed v. However, the direction of motion n共t兲 of each particle changes in time according to a rule that captures in a qualitative way the interactions between organisms in a flock. The basic idea is that each particle moves in the average direction of motion of the particles surrounding it, plus some noise. Two interaction rules have been considered in the literature, which differ in the way the noise is introduced into the system. To state these rules mathematically, we need some definitions. Let Rn共r兲 be the circular vicinity of radius r centered at xជ n共t兲, and Kn共t兲 be the number of particles whose positions are within Rn共r兲 at time t. We ជ 共t兲 the average velocity of the particles will denote as U n which at time t are within the vicinity Rn共r兲, namely,
ជ 共t兲 = U n
1 兺 vជ j共t兲. Kn共t兲 兵j:xជ j共t兲苸Rn共r兲其
共1兲
ជ 共t兲 the For reasons that will be clear later, we will call U n input signal received by the nth particle vជ n. With the above definitions, the interaction rule originally proposed by Vicsek et al. can be written as ជ 共t兲兴 + 共t兲, n共t + ⌬t兲 = Angle关U n n
共2a兲
vជ n共t + ⌬t兲 = vein共t+⌬t兲 ,
共2b兲
xជ n共t + ⌬t兲 = xជ n共t兲 + vជ n共t + ⌬t兲⌬t,
共2c兲
where n共t兲 is a random variable uniformly distributed in the interval 关− , 兴, and the noise amplitude is a parameter taking a constant value in 关0, 1兴. The “Angle” function is defined in such a way that if uជ = uei, then Angle 关uជ 兴 = . Note that in this case the direction of the neighbors’ average veជ 共t兲 共the input signal兲 is computed first and then the locity U n noise is added to this direction. We will refer to the interaction rule given in Eqs. 共2a兲–共2c兲 as the self-propelled model with intrinsic noise 共SPMIN兲, and to the term n共t兲 as the intrinsic noise. The second interaction rule, proposed by Grégoire and Chaté in Ref. 关12兴, is given by
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ជ 共t兲 + ein共t兲兴, n共t + ⌬t兲 = Angle关U n
共3a兲
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vជ n共t + ⌬t兲 = vein共t+⌬t兲 ,
共3b兲
xជ n共t + ⌬t兲 = xជ n共t兲 + vជ n共t + ⌬t兲⌬t,
共3c兲
where n共t兲 and have the same meaning as in Eq. 共2a兲. In this case, a random vector of constant length and random orientation n共t兲 is added to the neighbors’ average velocity ជ 共t兲, and then the direction of the resultant vector is comU n puted. We will refer to the model given in Eqs. 共3兲 as the self-propelled model with extrinsic noise 共SPMEN兲, and to the term ein共t兲 as the extrinsic noise. One might think that the two interaction rules 共2a兲 and 共3a兲 are more or less equivalent and should produce qualitatively similar dynamical behaviors. However, as mentioned in the Introduction, there is a clear physical difference between these two ways of adding noise to the system. In the SPMIN the uncertainty induced by the noise is in the decision mechanism 共the Angle function兲, but not in the input signal that each particle receives. In contrast, one could say that in the SPMEN the particles are “short-sighted” and do not see clearly the signal sent by the neighbors, which is what causes the uncertainty in this case. Thus, one cannot expect, a priori, the onset of collective order to be the same in both the SPMIN and the SPMEN, for the way in which the noise is introduced in both cases is clearly different, not only algorithmically, but also physically. To measure the amount of order in the system we define the instantaneous value of the order parameter 共t兲 as
共t兲 =
冏
N
冏
1 兺 vជ n共t兲 = v−1兩具Uជ 共t兲典兩, vN n=1
共4兲
ជ 共t兲典 = 1 兺N vជ 共t兲 is the average velocity of the entire where 具U N n=1 n system at time t. Thus, if 共t兲 ⬇ 0 the particles move in random uncorrelated directions, whereas if 共t兲 ⬇ 1, all the particles are aligned and move in the same direction. In the limit t → ⬁, the order parameter 共t兲 reaches a constant value = limt→⬁ 共t兲 that characterizes the steady state behavior of the system 关26兴. In their original work, Vicsek and his group noted the existence of a phase transition from ordered states 关共t兲 ⬎ 0兴 to disordered states 关共t兲 ⬇ 0兴 as the value of the noise intensity increases. We illustrate this phase transition for the SPMIN in Fig. 1共a兲, which was obtained numerically for a system with N = 20 000 particles within a box of sides L = 32. The interaction radius and particle speed used in the numerical simulation are r = 0.4 and v = 0.05, respectively. On the other hand, Fig. 1共b兲 shows the phase transition for the SPMEN with exactly the same parameters as in the previous case. As it can be seen from Fig. 1, the phase transition looks continuous 共second order兲 for the SPMIN, whereas it is clearly discontinuous 共first order兲 for the SPMEN. Numerical simulations performed for larger systems than the one used in Fig. 1 also seem to indicate that the phase transition exhibited in the SPMIN is continuous 关3–5兴. Nonetheless, based on numerical simulations, the authors of Ref. 关12兴 have pointed out that the phase transition may be discontinuous regardless of the type of noise, and that the apparent continuity of the phase transition in the SPMIN is due
FIG. 1. Phase transition in the self-propelled model with 共a兲 intrinsic noise and 共b兲 extrinsic noise. In the first case the phase transition seems to be continuous, whereas in the second case it is clearly discontinuous. The numerical simulations were performed using systems with N = 20 000, L = 32, r = 0.4, and v = 0.05.
to strong finite-size effects. For some reason, these finite size effects are not so strong in the SPMEN, in which the phase transition is clearly discontinuous. Due to a lack of a general mathematical formalism to analyze the self-propelled model 共either with intrinsic or extrinsic noise兲, the way in which each type of noise affects the dynamics of the system remains unknown. In what follows we present two simplified versions of the Vicsek model that can be solved analytically for the two types of noise. We show that in these simpler models, the phase transition is continuous for the intrinsic noise and discontinuous for the extrinsic noise. III. THE MAJORITY VOTER MODEL
The first network model that we consider consists of a set of N binary variables v1 , v2 , . . . , vN, each acquiring the values +1 or −1. The value of each vn changes in time and is determined by a set of kn other elements In = 兵vn1 , vn2 , . . . , vnk 其, which we will call the inputs of vn. n Thus, at every time step every element vn receives a signal from its set of inputs In and updates its value according to that signal. Different ways of assigning the inputs to each element lead to different network topologies. In this section we focus on the homogeneous random topology characterized by the following two properties. All the elements have the same number of inputs K, namely, kn = K for all n. The K inputs of each element are chosen randomly from anywhere in the system. Note that this is a directed network, for if vn is an input to vm, then vm is not necessarily an input to vn. We can think of this system as a society of N individuals in which every individual vn can have two opinions 共+1 and −1兲 about an issue. Each individual’s opinion is influenced by its K friends 共inputs兲, who are randomly chosen among the N individuals in the society. Generally, each individual will tend to be of the same opinion as the majority of its friends, but with a given probability it can have the opposite opinion. This probability of having an opinion opposite to that of the majority can be considered as a “temperature” that introduces noise in
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the dynamics of the system. Here we consider two ways of introducing this noise, which are analogous to the intrinsic and extrinsic noise in the self-propelled model. We define the input signal Un共t兲 influencing element vn共t兲 as K
Un共t兲 =
1 兺 vn 共t兲. K j=1 j
共5兲
This is just the average opinion of the inputs of vn. With this definition, the dynamics of the system with intrinsic noise is given by the simultaneous updating of the network elements according to the rule vn共t + 1兲 =
再
with prob. 1 − ,
sgn关Un共t兲兴
,
− sgn关Un共t兲兴 with prob.
冎
共6兲
where sgn关Un兴 = −1 if Un ⬍ 0 and sgn关Un兴 = 1 if Un ⬎ 0. If Un = 0 then we choose for vn共t + 1兲 either +1 or −1 with equal probability. The noise amplitude is a constant parameter in the interval 关0,1/2兴 that represents the probability for each individual to go against the majoritarian opinion. The above interaction rule can also be written in a simpler form as
冋
vn共t + 1兲 = sgn sgn关Un共t兲兴 +
册
n共t兲 , 1−
共7兲
where n共t兲 is a random variable uniformly distributed in the interval 关−1 , 1兴. From the above expression it is clear that this way of introducing the noise is equivalent to the intrinsic noise of Eq. 共2a兲, since the noise is added after the sgn function has been applied to the input signal Un共t兲. 共Since vn can only take the values +1 or −1, the sgn function has to be applied again.兲 The other way of introducing the noise, which is equivalent to the extrinsic noise in Eq. 共3a兲, is given by the interaction rule vn共t + 1兲 = sgn关Un共t兲 + 4n共t兲兴,
共8兲
where now the noise is directly added to the input signal Un共t兲 and then the sgn function is evaluated. n共t兲 and have the same meaning as in Eq. 共7兲. 共The factor 4 in the preceding equation is just to guarantee that the phase transition in this case occurs within the interval 苸 关0 , 1 / 2兴.兲 For both types of noise, intrinsic and extrinsic, the majority voter model exhibits a phase transition from ordered to disordered states. However, the nature of this phase transition 共i.e., whether continuous or discontinuous兲 depends on the type of noise. To see that this is indeed the case, we define the order parameter 共t兲 for the majority voter model as N
共t兲 =
1 兺 vn共t兲. N n=1
FIG. 2. Phase transition in the majority voter model with 共a兲 intrinsic noise and 共b兲 extrinsic noise. In the first case the phase transition appears to be continuous, whereas in the second case it is discontinuous. The symbols represent data obtained from numerical simulations using systems with N = 105 and K = 3. The solid lines correspond to the mean-field prediction.
1共a兲. Contrary to the above, the phase transition for the majority voter model with extrinsic noise 关Eq. 共8兲兴 is discontinuous, as is shown in Fig. 2共b兲, which is also consistent with the behavior observed in Fig. 1共b兲 for the SPMEN. Thus, changing the way in which the noise is introduced in the voter model also changes drastically the nature of the phase transition. The majority voter model is simple enough to be treated analytically. We can even generalize the model to incorporate the two types of noise simultaneously. In this generalization, the value of each element vn is updated according to the dynamical rule
冋
vn共t + 1兲 = sgn sgn关Un共t兲 + 41n共t兲兴 +
where n and n are independent random variables uniformly distributed in the interval 关−1 , 1兴, and 1 and 2 are constant parameters taking values in the interval 关0,1/2兴. Thus, if 1 = 0 and 2 ⫽ 0, only the intrinsic noise is present, whereas if 1 ⫽ 0 and 2 = 0 only the extrinsic noise is present. Intermediate cases are obtained if both 1 and 2 are different from zero. In what follows we consider separately the case in which K is finite, and the case in which K is infinite. A. Case 1: K ⬍ ⴥ
In Appendix A we present a mean-field calculation showing that, when the network connectivity K is finite, the order parameter 共t兲 satisfies the dynamical mapping
共9兲
In the limit t → ⬁, the order parameter 共t兲 reaches a stationary value that depends on the noise intensity . Figure 2共a兲 shows 兩兩 as a function of for the intrinsic noise case 关Eq. 共7兲兴, in a system with N = 105 and K = 3. It is apparent that in this case the phase transition is continuous. This result is consistent with the behavior of the SPMIN observed in Fig.
册
n共t兲 , 共10兲 1 − 2
共t + 1兲 = M关共t兲兴,
共11a兲
where K
K 共 1兲 m , M共兲 = 共1 − 22兲 兺 m m=1
K 共1兲 are given by and the coefficients m
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mK共1兲 =
冉冊
= M共兲 = 31共1兲 + 33共1兲3 .
K 共− i兲m−1 m 4K1
⫻
冕
⬁
关cos 兴K−m关sin 兴m
−⬁
sin共4K1兲 d. 2 共11c兲
In the calculations that lead to the set of equations 共11a兲 and 共11b兲 one assumes that the network elements vn are statistically independent and equivalent 共see Appendix A兲. These assumptions hold as long as the K inputs of each element are chosen randomly from anywhere in the system, namely, for the homogeneous random topology. For other topologies that introduce correlations between the network elements, such as the small-world or the scale-free topologies, the mean-field assumptions do not necessarily apply. However, when they do apply, the order parameter given in Eq. 共9兲 becomes the sum of N independent and equally distributed random variables. Therefore, the determination of 共t兲 becomes analogous to determining the average position of a onedimensional 共1D兲 biased random walk, which can be solved exactly. The stable fixed points of Eq. 共11a兲 give the stationary values = limt→⬁ 共t兲 of the order parameter. It is clear from Eqs. 共11a兲–共11c兲 that = 0 is always a fixed point. However, its stability depends on the values of 1 and 2. Additionally, K 共1兲 = 0 for even values of m from Eq. 共11c兲it follows that m 共because in such a case the integrand in that equation is an odd function兲. Therefore, the polynomial in Eq. 共11b兲 contains only odd powers of 共t兲 and thus, for each fixed point , the opposite value − is also a fixed point. To illustrate the formalism, we present here a detailed analysis of the simple case K = 3. However, the results are similar for any other finite value of K. For K = 3, the integrals in Eq. 共11c兲 can be easily computed 共we used MATHEMATICA 关23兴兲 and Eq. 共11b兲 becomes M共兲 = 共1 − 22兲关31共1兲 + 33共1兲3兴,
共12a兲
共13兲
Using Eqs. 共12b兲 and 共12c兲, it is easy to see that = 1 and = −1 are solutions of the fixed-point Eq. 共13兲 provided that 0 ⱕ 1 ⬍ 41 共in addition to the trivial solution = 0 which is always a fixed point兲. From Eqs. 共12b兲, 共12c兲, and 共13兲 we obtain that
M ⬘共1兲 = 31共兲 + 333共1兲 =
冦
if 0 ⱕ 1 ⱕ
0
1 , 12
121 − 1 1 1 ⱕ 1 ⱕ , if 12 4 81 1 41
if
1 ⱕ 1 . 4
冧
It follows from the above expression that 兩M ⬘共⫾1兲兩 ⬍ 1 in the region 0 ⱕ 1 ⬍ 41 , which shows that the fixed points = 1 and = −1 are stable in this region. For 1 ⬎ 41 the fixed points = ⫾ 1 disappear and the only fixed point that remains is = 0. Let us compute now the stability of the fixed point = 0. From Eqs. 共12b兲 and 共13兲 we get
M ⬘共0兲 = 31共兲 =
冦
3 2
if 0 ⱕ 1 ⱕ
1 , 12
1 1 121 + 1 if ⱕ 1 ⱕ , 12 4 161 1 41
if
1 ⱕ 1 4
冧
from which it follows that 兩M ⬘共0兲兩 ⬍ 1 for 1 ⬎ 41 , whereas 兩M ⬘共0兲兩 ⱖ 1 for 0 ⱕ 1 ⱕ 41 . The stability analysis presented above reveals that, when 2 = 0, the stable fixed points discontinuously transit from = ⫾ 1 to = 0 as 1 crosses the critical value c1 = 1 / 4 from below. Therefore, the phase transition in this case is discontinuous, as is shown in Fig. 2共b兲.
where
2. Subcase 2: 1 = 0
31共1兲 =
4 + 241 − 兩1 − 121兩 − 3兩1 − 41兩 , 321
33共1兲 = −
81 − 兩1 − 121兩 + 兩1 − 41兩 . 321
共12b兲
We now consider the case in which 1 = 0, that is, when only intrinsic noise is present. Taking the limit 1 → 0 in Eqs. 共12b兲 and 共12c兲 one gets 31共0兲 = 3 / 2 and 33共0兲 = −1 / 2. Therefore, in this case Eq. 共12a兲 becomes
共12c兲
To determine the stability of the fixed points of the mapping M共兲 we have to analyze the value of the derivative M ⬘共兲 ⬅ dM共兲 / d. If 兩M ⬘共兲兩 ⬍ 1 at the fixed point, then that fixed point is stable. Otherwise, it is unstable. We further divide our presentation in three cases. 1. Subcase 1: 2 = 0
Let us first show that the phase transition is discontinuous for the case in which 2 = 0, namely, when there is no intrinsic noise and only the extrinsic noise is present. Under these circumstances, the fixed-point equation = M共兲 becomes
M共兲 = 共1 − 22兲
冉
冊
1 3 − 3 . 2 2
共14兲
Let us start by analyzing the stability of the trivial fixed point = 0. From the above equation we get 3 M ⬘共0兲 = 共1 − 22兲 , 2 from which it follows that 兩M ⬘共0兲兩 ⬍ 1 only for 2 ⬎ 61 . Therefore, the disordered state characterized by the fixed point = 0 is stable only for 2 ⬎ 61 . As 2 decreases below the critical value c2 = 61 , the disordered phase becomes unstable and two stable nonzero fixed points appear. Assuming
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FIG. 3. 共Color online兲 Graph of the dynamical mapping M共兲 as a function of for K = 3. 共a兲 1 = 0.1 and different values of 2; 共b兲 2 = 0.1 and different values of 1. Note that in both cases, as the noise intensity decreases the stable nonzero fixed point appears continuously. This is always the behavior for finite values of K.
FIG. 4. 共Color online兲 Graph of the dynamical mapping M共兲 as a function of for K → ⬁. 共a兲 1 = 0.1 and different values of 2; 共b兲 2 = 0.1 and different values of 1. Note that in both cases, as the noise intensity decreases the stable nonzero fixed point appears discontinuously 共indicated by the arrows兲. For infinite K, the phase transition is always discontinuous.
= ⫾
⫽ 0, the fixed point equation = M共兲 can be solved for obtaining
冉
2 = ⫾ 3− 1 − 22
冊
1/2
.
A stability analysis reveals that the above fixed points are stable for 2 ⬍ c2 关in this region 兩M共兲兩 ⬍ 1兴 and unstable for 2 ⬎ c2 关because in this other region 兩M共兲兩 ⬎ 1兴. Summarizing, the stable fixed points for the case 1 = 0 are
冦
冉
2 ⫾ 3− 1 − 22 = 0
冊
1/2
if 0 ⱕ 2 ⱕ if c2 ⬍ 2 ,
c2 ,
冧
where c2 = 1 / 6. This result is plotted in Fig. 2共a兲 共solid line兲, from which it is apparent that the phase transition in the majority voter model with only intrinsic noise is indeed continuous. Additionally, for values of 2 below, but close to, the critical value c2 at which the phase transition occurs, the order parameter behaves as ⬇ ⫾ 共c2 − 兲1/2, which shows that this phase transition belongs to the mean-field universality class. 3. Subcase 3: 1 Å 0 and 2 Å 0
When both types of noise, intrinsic and extrinsic, are present in the system, the phase transition is always continuous for any finite value of K. To illustrate this we present in Fig. 3共a兲 the graph of M共兲 for 1 = 0.1 and different values of 2. Note that M共兲 is a monotonically increasing convex function, and therefore the nonzero stable fixed point appears continuously as 2 decreases. The same happens if we now fix the value of 2 and vary the value of 1, as it is shown in Fig. 3共b兲. This behavior is typical of a continuous phase transition. Assuming ⫽ 0 and using Eq. 共12a兲, the fixed point equation = M共兲 can be solved for obtaining
再 冉 1
1 − 31共1兲 3  3共 1兲 1 − 2 2
冊冎
1/2
共15兲
For this equation to have real solutions the quantity inside the curly brackets must be positive. From Eq. 共12c兲 it follows that 33共1兲 ⱕ 0 for any positive value of 1. Therefore, Eq. 共15兲 has real solutions only if 1 ⱕ 31共1兲. 1 − 22 The values of 1 and 2 for which the equality holds in the above expression determine the critical line on the 1-2 plane at which the phase transition occurs. Figure 5 shows surface plots for the 共positive兲 value of the stable fixed point as a function of 1 and 2 for K = 3, K = 9, K = 15, and K → ⬁. Interestingly, for any finite value of K the phase transition is always continuous except for the special case 2 = 0. Therefore, for any finite K, even a small amount of intrinsic noise suffices to make the phase transition continuous. B. Case 2: K \ ⴥ
In Appendix A we show that for K → ⬁ the temporal evolution of the order parameter is still given by the dynamical mapping Eq. 共11a兲, where now M共兲 is
M共兲 =
冦
− 共1 − 22兲 if ⬍ − 41 , 1 − 22 41
if 兩兩 ⱕ 41 ,
1 − 22
if ⬎ 41 .
冧
共16兲
Figure 4共a兲 shows the behavior of M共兲 for K → ⬁, 1 = 0.1 and different values of 2, and Fig. 4共b兲 shows the same kind of plots but now keeping 2 = 0.1 and varying the value of 1. It can be seen from Fig. 4共a兲 that the phase transition is discontinuous. Indeed, as 2 decreases below the critical value c2 = 0.3, the nonzero stable fixed point appears discon-
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ties 1 and 2, which take constant values between 0 and 1, are the amplitudes of the extrinsic 共Grégoire-Chaté兲 and intrinsic 共Vicsek兲 types of noise, respectively. Note that, while in the self-propelled model the particles can move and thus the vectors vជ n represent particle velocities, in the VNM the particles do not move. Rather, they are fixed to the nodes of the network. For this reason, in the VNM the vectors vជ n cannot be considered as velocities, but just as a given property of the particles 共such as spin兲. A. Equivalence between the VNM and the self-propelled model FIG. 5. Phase transition in the majority voter model. Order parameter as a function of the extrinsic noise 1 and the intrinsic noise 2 for K = 3, K = 9, K = 15 and K → ⬁. Note that for any finite value of K the phase transition is always continuous except for the case 1 = 0. Note also that the phase transition it is always discontinuous for K → ⬁.
tinuously 共see the point indicated with an arrow in the figure兲. An analogous behavior occurs in Fig. 4共b兲 when 1 reaches the value c1 = 0.2. Thus, in the limit K → ⬁ the phase transition is always discontinuous 共see Fig. 5兲. In this sense, the discontinuity in the phase transition observed when only extrinsic noise is used can be considered as a singular limit, either 2 → 0 or K → ⬁, of a phase transition that is otherwise continuous. IV. THE VECTORIAL NETWORK MODEL
The second network model that we analyze, which we will call the “vectorial network model” 共VNM兲, is much closer to the self-propelled model than the voter model presented in the previous section. As we will see later, the VNM corresponds to a mean-field theory of the self-propelled model. It consists of a network with N nodes 共or elements兲 which, as in the self-propelled model, are the twodimensional vectors vជ 1 = ei1 , . . . , vជ N = eiN 共represented as complex numbers兲. All the vectors have the same magnitude 兩vជ n兩 = 1 but their orientations 1 , . . . N in the plane can change. Each vector vជ n is connected to a fixed set of kn other vectors, In = 兵vជ n1 , vជ n2 , . . . , vជ nk 其, from which vជ n will receive an n input signal. We will call this set the inputs of vជ n, and consider again the homogeneous random topology in which all the elements have exactly K inputs chosen randomly from ជ 共t兲 received by anywhere in the system. The input signal U n vជ n from its K inputs is defined as K
ជ 共t兲 = 1 兺 vជ 共t兲. U n n K j=1 j
共17兲
For the interaction between the network elements we consider from the beginning a dynamic rule that already incorporates both types of noise, intrinsic and extrinsic:
ជ 共t兲 + ein共t兲兴 + 共t兲, n共t + 1兲 = Angle关U n 1 2 n
共18兲
where n共t兲 and n共t兲 are independent random variables uniformly distributed in the interval 关− , 兴. The noise intensi-
The main difference between the self-propelled model and the VNM is that in the former the motion of the particles can produce correlations in space and time that may couple the global order with the local density, affecting the phase transition. Such coupling is not present in the VNM unless we choose very specific network topologies that change over time. However, here we are interested only on the effects that the two different types of noise have on the phase transition. The VNM is especially suited for this analysis precisely because of the absence of such complicated dynamical effects as the coupling between order and density. Nonetheless, it is important to note that the limit of large particle speeds of the self-propelled model is well described by the VNM. Indeed, if the speed of the particles in the self-propelled model is small enough, then particles that were within the same interaction vicinity at time t will most likely remain within the same interaction vicinity at the next time step t + ⌬t. Therefore, for small particle speeds the spatial correlations between the particles are important. Contrary to this, in the opposite limit of large particle speeds, and because of the noise in the direction of motion of each particle 共whether intrinsic or extrinsic兲, particles that at time t were within the same vicinity will most likely not remain within that vicinity and end up interacting with different particles at the next time step t + ⌬t. Therefore, in the limit v → ⬁ of the self-propelled model spatial correlations are lost in only one time step. More precisely, the particle speeds must be great enough to separate nearby particles distances larger than any correlation length in the system at each time step. This, then, is essentially equivalent to randomly mixing the particles within the box at every time step, as was done to obtain Figs. 6 and 7, and is the condition for the mean-field theory conveyed in the VNM to be applicable. Figure 6 shows the value of the order parameter as a function of the noise intensity for the VNM with intrinsic noise only 共solid line兲, and for the SPMIN with random mixing 共symbols兲 关27兴. Namely, instead of updating the positions of the particles in the SPMIN according to the kinematic rule given in Eq. 共2c兲, we just randomly mixed all the particles within the box at every time step. We used equivalent systems with N = 20 000 particles and adjusted the other parameters of the SPMIN in such a way that the average number of interactions per particle K was the same as for the vectorial network model 关K = 5 in Fig. 6共a兲 and K = 20 in Fig. 6共b兲兴兲. Analogously, Fig. 7 shows equivalent results but for the VNM with extrinsic noise only 共solid line兲 and the SPMEN
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FIG. 6. Phase transition for the VNM with intrinsic noise only 共solid line兲 and the SPMIN with random mixing 共symbols兲, both with N = 20 000. For the SPMIN the size of the box is L = 32 and the radius r of the interaction vicinity has been chosen so that the average number of interactions per particle coincides with the network connectivity K in the VNM. The two panels correspond to 共a兲 K = 5 共r ⬇ 0.285兲 and 共b兲 K = 20 共r ⬇ 0.571兲. Note that the phase transition in this case is continuous and the same for both the VNM and the SPMIN with random mixing.
with random mixing 共symbols兲. As we can see from these figures, regardless of the type of noise both the self-propelled model with random mixing and the VNM give the same phase transition within numerical accuracy. Although the above results do not constitute a proof, they do suggest that, in the limit of large particle speeds, the self-propelled model becomes equivalent to the VNM. B. Mean-field theory of the VNM
value of the order parameter 共t兲 is related to this vector ជ 共t兲兩. Let 关共t兲 , 共t兲兴 be the polar coordinates through 共t兲 = 兩U ជ of U共t兲, and PUជ 共 , ; t兲 its probability distribution function 共in polar coordinates兲. Note, then, that 共t兲 is the first radial moment of PUជ 共 , ; t兲. As in the voter model, we assume that the network elements vជ n are statistically independent and equivalent. Therefore, Eq. 共19兲 becomes the sum of N independent and equally distributed random variables, and the problem of determinជ 共t兲 is then similar to that of finding the position of a 2D ing U biased random walk. In Appendix B we present this meanfield calculation, and show that the temporal evolution of the Fourier transform of PUជ 共 , ; t兲 is given by the recurrence relation ⬁
PˆUជ 共, ␥ ;t + 1兲 = J0共兲 + ⫻
⬁
0
⫻
2
sin共m2兲 Jm共兲eim␥ 2 2 2
d⬘ J0共K1⬘兲 ⬘ d␥⬘关PˆUជ 共⬘, ␥⬘ ;t兲兴Ke−im␥⬘ ,
共20兲
0
where the Jm共x兲’s are Bessel functions, and and ␥ are the Fourier conjugate variables to and , respectively. In principle, this equation can be solved for any finite value of the network connectivity K. However, we were able to solve it only in the limit case K → ⬁. Therefore, we present first the analytic results for K → ⬁. In the next section we present numerical results for finite values of K.
ជ 共t兲 as Let us define the average vector U
C. Case 1: K \ ⴥ
N
ជ 共t兲 = 1 兺 vជ 共t兲. U n N n=1
冕 冕
兺 m=−⬁
共19兲
ជ 共t兲 is the average In the context of the self-propelled model, U velocity over the entire system. Clearly, the instantaneous
In the limit K → ⬁ the factor 关PˆUជ 共⬘ , ␥⬘ ; t兲兴K appearing in the second integral of Eq. 共20兲 can be replaced by a Dirac ␦ function radially centered at K共t兲. This leads to ⬁
PˆUជ 共, ␥ ;t + 1兲 = J0共兲 +
共− i兲m 兺 m=−⬁
⫻ eim共␥−␣兲
冕
⬁
0
sin共m2兲 Jm共兲 2
dx J0共x兲Jm关共t兲x兴. x
共21兲
From this equation it follows that the order parameter 共t兲, which is the first radial moment of PUជ 共 , ; t兲, obeys the dynamical mapping 共see details in Appendix B兲
共t + 1兲 =
FIG. 7. Phase transition for the VNM with extrinsic noise only 共solid line兲 and the SPMEN with random mixing 共symbols兲. The systems have the same parameters as in Fig. 6: N = 2 ⫻ 104, L = 32, K = 4 共r ⬇ 0.285兲 in 共a兲 and K = 20 共r ⬇ 0.571兲 in 共b兲. Note that the phase transition in this case is discontinuous and the same for both the VNM and the SPMEN with random mixing.
sin共2兲 2
冕
⬁
J0共1x兲J1关共t兲x兴
0
dx . x
共22兲
The integral on the right-hand side of the above equation is an instance of the Weber-Schafheitlin integrals 关24兴. After the evaluation of this integral, the dynamical mapping for the order parameter can be written as
共t + 1兲 = M关共t兲兴, where the mapping M共兲 is
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FIG. 8. Phase transition in the VNM with K → ⬁. The left panel corresponds to the phase transition obtained starting out the dynamics from a fully disordered condition 关共0兲 ⬇ 0兴, whereas for the panel on the right the dynamics were started from fully ordered initial conditions 关共0兲 ⬇ 1兴. In both cases the phase transition is discontinuous for any nonzero value of the extrinsic noise amplitude 1. These plots were obtained by numerically finding the fixed points of M共兲 given in Eq. 共22兲
M共兲 =
冦
冉
冋 册冊 冋 册冊
sin共2兲共t兲 1 1 , ;2, 2F 1 2 2 2 1 2 1
冉
1 1 sin共2兲 1 ,− ;1, 2F 1 2 2 2
2
2
if ⬍ 1 , if ⬎ 1
冧
共23b兲
and 2F1共a , b ; c , d兲 are hypergeometric functions. The fixed points of M共兲 give the stationary value of the order parameter. In Ref. 关14兴 we have shown that for 2 = 0, the nontrivial fixed point of this mapping appears discontinuously as 1 crosses the critical value 1 ⬇ 0.672 from above. On the sin共 兲 other hand, for 2 ⬎ 0 there is a global factor 2 2 which does not change the discontinuous appearance of the nontrivial fixed point. Therefore, for any values of 1 and 2, the phase transition is discontinuous. Figure 8 shows surface plots of the stationary value as a function of 1 and 2 for two different cases: 共i兲 when the dynamics of the VNM start out from disordered initial conditions 关共0兲 ⬇ 0 in Eq. 共23a兲兴 and 共ii兲 when the dynamics start out from ordered initial conditions 关共0兲 ⬇ 1 in Eq. 共23a兲兴. Let us denote the stationary values of the order parameter obtained in each of the two cases mentioned above as dis and ord, respectively. It is apparent from Fig. 8 that dis and ord are equal in a large region of the 1-2 parameter space. However, there is also a region in which dis and ord are different. This latter region, where the system shows hysteresis, is shown in Fig. 9, in which the difference ⌬ = ord − dis is plotted as a function of 1 and 2. The region for which ⌬ ⫽ 0 is a region of metastability where two stable fixed points exist, the trivial one dis = 0 and the nonzero fixed point ord. It is clear from these results that the VNM with K → ⬁ exhibits a discontinuous phase transition for any nonzero value of the extrinsic noise 1. On the other hand, for 1 = 0, the amount of order in the system decreases as 2 increases. However, there is no phase transition in this case since = 0 only when 2 reaches its maximum value 2 = 1. 共In ferromagnetic systems, 2 = 1 would correspond to infinite temperature.兲 In other words, for infinite connectivity and zero extrinsic noise, the order in the system can never be destroyed by the intrinsic noise, unless it reaches its maximum value. However, in the presence of both types of noise,
FIG. 9. Difference ⌬ = ord − dis between the two surface plots displayed in Fig. 8. The region for which ⌬ ⫽ 0 is the region of metastability where the system exhibits hysteresis. Note that this region crosses the entire square 关0 , 1兴 ⫻ 关0 , 1兴 in the 1-2 plane from one side to the other.
extrinsic and intrinsic, the phase transition is always discontinuous. D. Case 2: K finite
As it was mentioned before, we do not have an analytic solution of Eq. 共20兲 for finite values of K. Nonetheless, numerical simulations show a phase transition that is continuous in one region of the 1-2 parameter space, and discontinuous in another region. This is qualitatively different from the phase transition observed in the majority voter model, which was always continuous for any finite value of K 共except for 2 = 0兲. Figure 10 shows as a function of 1 and 2 for different values of K. The results reported in this figure were obtained through numerical simulations of the VNM for systems with N = 20 000. The figures on the left correspond to disordered initial conditions 关共0兲 ⬇ 0兴, whereas those on the right correspond to ordered initial conditions 关共0兲 ⬇ 1兴. The difference ⌬ = ord − dis as a function of 1 and 2 is plotted in Fig. 11. It is apparent from these figures that, except for the case K = 3, there is a region of hysteresis where ⌬ ⫽ 0. This region grows as K increases, but it does not seem to cross the square 关0 , 1兴 ⫻ 关0 , 1兴 of the 1-2 parameter space from one side to the other 共as it does for K → ⬁兲. This implies that the phase transition is discontinuous along the boundary of the region in which ⌬ ⫽ 0, but it is continuous along the boundary where → 0+ and ⌬ = 0. It is worth emphasizing that for 1 = 0, namely, when there is no extrinsic noise, the VNM always undergoes a continuous phase transition from ordered to disordered states as the intensity of the intrinsic noise 2 increases. This is consistent with the behavior originally reported by Vicsek et al. for the SPMIN 关see Fig. 1共a兲兴. On the other hand, for 2 = 0, i.e., in the absence of intrinsic noise, the phase transition in the VNM is discontinuous as a function of the extrinsic noise amplitude 1, which is consistent with the phase transition observed in the SPMEN 关see Fig. 1共b兲兴. V. WHAT DO WE MEAN BY “MEAN-FIELD” THEORY?
In the computations of these two network models we have used the “mean field” assumption that all the network ele-
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FIG. 10. Surface plots show the order parameter as a function of the extrinsic and extrinsic noise amplitudes 1 and 2 respectively, for K = 3, K = 9 and K = 15. The graphs were obtained through numerical simulation for systems with K = 2000 elements. The panels on the left correspond to random initial conditions, whereas the panels on the right correspond to fully ordered initial conditions. Note that, except for the case K = 3, there is a region of hysteresis where the phase transition is discontinuous.
ments are statistically independent and statistically equivalent, though we consider arbitrary values of K 共the number of particles that interact兲. This leads to phase transitions that can be continuous or discontinuous depending upon the values of 1 and 2. However, a frequent 共and frequently equivalent兲 view of what constitutes a “mean field” theory entails the assumption that every particle interacts with all the other particles in the system. This second assumption is akin to the case K → ⬁, for which the phase transition is always discontinuous and, in general, of a different nature than the one obtained for finite K. In our network models, the assumption of statistical independence is certainly true for the homogeneous random topology in which the K inputs of each element are randomly chosen from anywhere in the system. In this case, the probability for two distinct elements vជ m and vជ n to have at least one of its K inputs in common is of order 1 / N. In the thermodynamic limit N → ⬁ this probability is zero. Therefore, for large systems all the elements have different sets of inputs and are indeed statistically independent. For other topologies, such as small-world, a large fraction of the network elements share inputs even in the thermodynamic limit. For this topology the assumption of statistical independence is no longer valid. However, the qualitative behavior of the phase transition obtained for the VNM on small-world networks, in which the short-range interactions induce spatial correlations between the network elements, is similar to the one observed for the homogeneous random topology described so far. Figure 12 shows the phase transition in the VNM on small-world networks. To generate this
FIG. 11. Difference ⌬ = ord − dis between the surface plots displayed in Fig. 10. The region of metastability where ⌬ ⫽ 0 grows with K. However, for any finite value of K this region does not cross the entire square 关0 , 1兴 ⫻ 关0 , 1兴 in the 1-2 plane.
figure we used the standard Watts-Strogatz small-world algorithm 关16–18兴, placing the elements on a 2D square lattice. Initially each element vជ n has K = 5 inputs, which are chosen as its four first-neighbors and the element vជ n itself 共i.e., we allow self-interactions兲. Then, with probability p each input connection in the network is rewired to a randomly chosen element. Thus, if p = 0 we have a regular square lattice, whereas if p = 1 the topology becomes homogeneously random. This latter case is where our mean-field theory results are exactly applicable. Figure 12共a兲 corresponds to the case in which there is no intrinsic noise 共2 = 0兲 and only the extrinsic noise is present, and Fig. 12共b兲 shows the opposite case where there is no extrinsic noise 共1 = 0兲. The different curves in each figure correspond to different values of the rewiring probability p. Note that the nature of phase transition, i.e., whether continuous or discontinuous, does not change with the rewiring probability p. What changes is the critical value of the noise at which the phase transition occurs, but the continuity of the phase transition does not change with p. This is important because for small values of p there are strong spatial correlations in the system generated by the first-neighbor interactions in the small-world network. However, even in the pres-
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FIG. 12. 共Color online兲 Phase transition in the VNM on smallworld networks. 共a兲 In the absence of intrinsic noise, the extrinsic noise produces a discontinuous phase transition regardless of the rewiring connectivity p. 共b兲 The phase transition is continuous for all values of p when only the intrinsic noise is present.
ence of such spatial correlations, the phase transition appears to be discontinuous for the extrinsic noise and continuous for the intrinsic noise. One advantage of the VNM is that the finite-size effects are much smaller than in the self-propelled model. Therefore, the continuous or discontinuous character of the phase transition can be very well observed with N = 20 000 particles. VI. SUMMARY AND DISCUSSION
Although the self-propelled model proposed by Vicsek et al. is one of the simplest models that we have to describe the emergence of collective order in groups of organisms, its analytic solution remains an open problem. Indeed, due to the lack of an analytical approach to analyze the dynamical properties of the model, it has not been possible to characterize unambiguously some of its most basic features. In this work we have been particularly interested in how the way in which the noise is introduced in the system may affect the nature of the phase transition. The original numerical simulations of the self-propelled model with intrinsic noise proposed by Vicsek and his group, seem to indicate that the system undergoes a continuous phase transition 关2兴. In contrast, the extrinsic noise later introduced by Grégoire and Chaté generates a phase transition that is clearly discontinuous 关12兴. To determine the effect that each type of noise can have on the phase transition, we have presented two network models that capture some of the main characteristics of the interactions in the self-propelled model and that can be handled analytically. These two network models, which we call the majority voter model and the vectorial network model, can be considered as mean-field representations of the original self-propelled model. In fact, we have shown numerically that the self-propelled model with random mixing, which is obtained in the limit of very large particle speeds, is indeed equivalent to the vectorial network model. When the number of interactions per particle is finite, our numeric and analytic results for the two network models
show that, in the absence of extrinsic noise, the phase transition driven by the intrinsic noise is continuous, in accordance with the interpretation reported for the original Vicsek model 共SPMIN兲 in Ref. 关2兴. While our results do not validate Vicsek’s original interpretation, the consistency with it is encouraging in that it might help to find out the causes, other than the type of noise, by which the phase transition may not be of second order in the SPMIN. In the opposite case, namely, when there is no intrinsic noise and only the extrinsic noise is present, the phase transition is discontinuous, which is consistent with the behavior reported by Grégoire and Chaté for the SPMEN 关12兴. For intermediate cases in which both types of noise are present, and for finite network connectivities, the situation is more complicated: The voter model exhibits a phase transition that is always continuous, whereas for the vectorial model there is a region in the noise-parameter space where the phase transition is continuous, and another region where it is discontinuous. In this later case, the region of discontinuity grows as the network connectivity increases. In the limit of infinite network connectivities and in the presence of both types of noise, the phase transition is always discontinuous in both the voter model and the vectorial model. Furthermore, numerical simulations indicate that our mean-field results appear to hold qualitatively also for the VNM with small-world topology. For this topology the mean-field assumptions are not strictly valid due to the existence of strong spatial correlations induced by the firstneighbor connections. An interesting question that arises from these results concerns the actual behavior of self-propelled models under the influence of both types of noise. While our results may describe adequately the high velocity regimes, at lower velocities spatial correlations may build up giving rise to different phenomena. Yet, even the determination of the nature of the phase transition for pure intrinsic noise has been subject of controversy 关5,12,13兴, so a detailed unambiguous numerical survey of the effects of both noises on self propelled models, at various velocities and densities, is a rather formidable task. However, such a survey could help determine to what extent can the stationary collective features of the network systems agree quantitatively with those of the corresponding swarming models. It has been suggested that the coupling between global order and local density could make the phase transition in the self-propelled model discontinuous regardless of the type of noise 关13兴. However, it is not clear how such coupling intervenes in the onset of collective order in the self-propelled model, nor how it can change the phase transition from continuous to discontinuous. In this direction, a model similar to the Vicsek model with intrinsic noise but with binary interactions only, has been reported in Ref. 关25兴. In that work, the authors find an apparently continuous phase transition, which is destabilized by long-wavelength density perturbations. The authors remark that an instability of this type might be behind the wavy behavior frequently observed in simulations of self-propelled models. Whether this instability survives or not when many simultaneous interactions are present, this kind of effects are presently beyond the scope of simple mean-field descriptions of swarming models. Clearly, to understand the actual behavior of swarming sys-
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tems and to determine to what extent the results obtained for the network models presented here can actually be applied to the self-propelled model, further analysis is still required. ACKNOWLEDGMENTS
This work was supported by the National Science Foundation under Grant No. DMS-0507745 共C.H.兲, by CONACyT Grant No. P47836-F and by PAPIIT-UNAM Grant No. IN112407-3 共M.A. and H.L.兲. J.A.P. acknowledges CONACyT for support. APPENDIX A: ANALYTIC RESULTS FOR THE MAJORITY VOTER MODEL
In this appendix we present the computation leading to Eqs. 共11a兲–共11c兲 and 共16兲. Let us start by defining the quantities
n共t兲 = 4K1n共t兲, un共t兲 = 兺 vn j共t兲,
共A1b兲
sn共t兲 = un共t兲 + n共t兲.
共A1c兲
j=1
With these definitions, the dynamical interaction rule Eq. 共10兲 can be rewritten in the equivalent form vn共t + 1兲 =
再
with prob. 1 − 2 ,
− sgn关sn共t兲兴 with prob.
2 .
冎
共A2兲
Ps共x,t兲 = 关Pu ⴱ P兴共x,t兲,
共A3兲
The first term on the right-hand side of the above equation accounts for the case in which sn共t兲 ⬎ 0 and the sgn function is evaluated. The second term takes into account the case in which sn共t兲 ⬍ 0 and the ⫺sgn function is evaluated. Now, we introduce the mean-field assumptions used in this computation: 共a兲 The probabilities n共t兲 and P+n 共t兲 are the same for all the elements in the network 共i.e., they are statistically equivalent兲. Thus, from now on we will drop off the subscript “n” in these quantities and use the notation n共t兲 = 共t兲 and P+n 共t兲 = P+共t兲. 共b兲 The network elements vn are statistically independent. This implies that the sum un共t兲 defined in Eq. 共A1b兲 can be considered as the sum of K independent and identically distributed random variables. Note that the order parameter 共t兲 = 具vn共t兲典 can be expressed in terms the probability 共t兲 as 共t兲 = 具vn共t兲典 = 共1兲共t兲 + 共−1兲关1 − 共t兲兴 = 2共t兲 − 1, from which we obtain 1 + 共t兲 共t兲 = . 2
共A6兲
where “ⴱ” denotes a convolution. Analogously, since un共t兲 is the sum of the K-independent random variables vn1共t兲 , . . . , vnK共t兲, all equally distributed with the PDF Pv共x , t兲, then Pu共x , t兲 is the K-fold convolution of Pv共x , t兲 with itself, and therefore, Ps共x , t兲 becomes
Ps共x,t兲 = 关Pv ⴱ ¯ ⴱ Pv ⴱ P兴共x,t兲.
共A7兲
K times
It is convenient to transform the above equation to Fourier space 共in the variable x兲. Denoting as Pˆv共 , t兲, Pˆu共 , t兲, Pˆs共 , t兲, and Pˆ共兲 the Fourier transforms of the corresponding PDF’s, we obtain Pˆs共,t兲 = 关Pˆv共,t兲兴K Pˆ共,t兲.
共A8兲
We have now to consider two cases separately, the case for which K ⬍ ⬁, and the case K → ⬁. 1. Case 1: K ⬍ ⴥ
Let n共t兲 and P+n 共t兲 be the probabilities that vn共t兲 = 1 and sn共t兲 ⬎ 0, respectively. From Eq. 共A2兲 it is clear that n共t兲 and P+n 共t兲 are related through
n共t兲 = 共1 − 2兲P+n 共t兲 + 关1 − P+n 共t兲兴2 .
共A5兲
The next step in this calculation consists in expressing P+共t兲 as a function of 共t兲, or equivalently, as a function of 共t兲. To this end, let Pv共x , t兲, Pu共x , t兲, Ps共x , t兲, and P共x兲 be the probability density functions of the random variables vn共t兲, un共t兲, sn共t兲, and n共t兲, respectively. 关Note that P共x兲, the probability density function of the noise, is independent of time.兴 From Eq. 共A1兲 it follows that
共A1a兲
K
sgn关sn共t兲兴
共t + 1兲 = 2共1 − 22兲P+共t兲 + 22 − 1.
By definition of 共t兲, we have that Pv共x,t兲 = 共t兲␦共x − 1兲 + 关1 − 共t兲兴␦共x + 1兲, where ␦共x兲 is the Dirac delta function. Therefore 关28兴, Pˆv共,t兲 = 共t兲e−i + 关1 − 共t兲兴ei . The above equation can be written in terms of 共t兲 by using the relationship given in Eq. 共A4兲, which gives Pˆv共,t兲 = cos − i共t兲sin .
共A9兲
On the other hand, n共t兲 is a random variable uniformly distributed in the interval 关−4K1 , 4K1兴, from which it follows that Pˆ共兲 =
1 sin 4K1. 4K1
共A10兲
Substituting into Eq. 共A8兲 the results given in Eqs. 共A9兲 and 共A10兲 we obtain
共A4兲
Equation 共A3兲 can then be written in terms of the order parameter 共t兲 as 061138-12
sin共4K1兲 Pˆs共,t兲 = 关cos − i共t兲sin 兴K 4K1 K
=
兺 共− i兲m m=0 ⫻
冉冊
K 关cos 兴K−m关共t兲sin 兴m m
sin共4K1兲 . 4K1
共A11兲
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INTRINSIC AND EXTRINSIC NOISE EFFECTS ON ...
age 共t兲 and variance 1 − 关共t兲兴2. Therefore, for very large values of K the central limit theorem allows us to approximate Pu共x , t兲 by a Gaussian with average K共t兲 and variance 2 = K兵1 − 关共t兲兴2其
Since P+共t兲 is the probability that sn共t兲 ⬎ 0, then
冕
P+共t兲 =
⬁
Ps共x,t兲dx.
0
Therefore, taking the inverse Fourier transform of Eq. 共A11兲 and integrating the result from 0 to ⬁ we obtain
Pu共x,t兲 ⬇
K
1 K 共1兲关共t兲兴m , P 共t兲 = 兺 m 2 m=0
共A12兲
+
冉冊
K 共− i兲m m 4K1
冕冕 ⬁
关cos 兴K−m关sin 兴m
⫻ 共A13兲
Although it is not obvious from the above expression, it happens that K0 共1兲 = 1. To show that this is indeed the case, let ˆ 共兲 as us define the function G
P+共t兲 =
冕 冕 1 2
0
⬁
ˆ 共兲eixddx = 2 G
−⬁
冕
冉冊 冕
冉
4K1
exp −
−4K1
G共x兲dx,
P+共t兲 =
冊
关x − y − K共t兲兴2 dydx. 2K兵1 − 关共t兲兴2其
8 1冑 ⫻
1 2 K 兵1
冕冕 ⬁
K→⬁
共A14兲
K0 共1兲 = 1,
冉
冊
K
Finally, substituting the above result into Eq. 共A5兲 we obtain
共t + 1兲 = 共1 − 22兲 兺
冊
− 关共t兲兴2其
冊
dy ⬘dx⬘ .
= ␦关x⬘ − y ⬘ − 共t兲兴,
1 81
冕冕 ⬁
0
41
−41
␦关x⬘ − y ⬘ − 共t兲兴dx⬘dy ⬘ .
After performing the integrals in the above expression we obtain
共A15兲
P+共t兲 =
m=1
2. Case 2: K \ ⴥ
By definition 关see Eq. 共A1b兲兴 un共t兲 is the sum of K independent and identically distributed variables, each with aver-
关x⬘−y ⬘−共t兲兴2 2 兵1−关共t兲兴2其 K
2 K 兵1
冑 2K 兵1 − 关共t兲兴2其
P+共t兲 =
K
mK共1兲关共t兲兴m .
关x⬘ − y ⬘ − 共t兲兴2
exp −
where ␦共¯兲 is the Dirac delta function. From the last two equations above it follows that, in the limit K → ⬁, the probability P+共t兲 acquires the simpler form
Eq. 共A12兲 can be written as
1 K 1 + 兺 m 共1兲关共t兲兴m . P+共t兲 = 2 m=1
冉
41
In the limit K → ⬁ we have
lim sin共4K1兲 d. 2
− 关共t兲兴2其
−41
0
exp −
关cos 兴K−m关sin 兴m
冉
冕冕 ⬁
0
−⬁
Using the fact that
冊
关x − y − K共t兲兴2 dy, 2K兵1 − 关共t兲兴2其
1
m−1
⬁
−4K1
0
K 共− i兲 m 4K1
⫻
exp −
Performing the change of variable x = Kx⬘, y = Ky ⬘, the above expression transforms into
⬁
ˆ 共兲. Since where G共x兲 is the inverse Fourier transform of G ˆ 共兲 is symmetric, then G共x兲 is also symmetric and therefore G ⬁ ˆ 共0兲 = 1 then G共x兲dx = 2兰⬁0 G共x兲dx. Additionally, since G 兰−⬁ ⬁ K 兰−⬁G共x兲dx = 1, from which it follows that 0 共1兲 = 1. For m ⱖ 1 we can exchange the order of integration in Eq. 共A13兲 by multiplying the integrand by e−x. After performing the integral over x and then taking the limit ⑀ → 0 we obtain
mK共1兲 =
冕
冉
4K1
8K1冑2K兵1 − 关共t兲兴2其 ⫻
ˆ 共兲 is a symmetric function and that G ˆ 共0兲 = 1. Note that G K With this definition, the coefficient 0 共1兲 can be written as ⬁
冑2K兵1 − 关共t兲兴 其
where we have used the fact that P共x兲 is a constant normalized function defined in the interval x 苸 关−4K1 , 4K1兴. From the above expression we can obtain P+共t兲 by integrating Ps共x , t兲 from 0 to ⬁:
ˆ 共兲 = 关cos 兴K sin共4K1兲 . G 4K1
K0 共1兲 = 2
兲.
2
8K1冑2K兵1 − 关共t兲兴2其
⬁
sin共4K1兲 ix e ddx.
⫻
2K兵1−关共t兲兴2其
1
Ps共x,t兲 =
−⬁
0
关x−K共t兲兴2
With this approximation, Eq. 共A6兲 becomes
K where the coefficients m 共1兲 are given by
mK共1兲 =
共
exp −
冦
0
if 共t兲 ⬍ − 41 ,
1 关共t兲 + 42兴 if 兩共t兲兩 ⱕ 41 , 81 1
if 共t兲 ⬎ 41 .
冧
This last result combined with Eq. 共A5兲 gives Eq. 共16兲 of the main text.
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PHYSICAL REVIEW E 77, 061138 共2008兲
TABLE I. Notation guide for the different quantities involved in the calculation of the phase transition of the VNM. We have omitted the subscript m in the PDF’s since we assume that all the network elements vជ m are statistically equivalent.
1. Order parameter
To measure the amount of order in the system, we define the instantaneous order parameter 共t兲 as
Amplitude
Scalar ni = 2共t兲 vector
polar coor.
PDF P ni共 兲
Fourier transform pˆm
nជ e共t兲 = K1ei共t兲
共r , 兲
Pnជ e共r , 兲
vជ m共t兲 = eim共t兲
共v , 兲
Pvជ 共v , ; t兲
Pˆnជ e共 , ␥兲 Pˆ ជ 共 , ␥ ; t兲
uជ m共t兲 = 兺Kj=1vជ m j共t兲
共u , 兲
Puជ 共u ,  ; t兲
ជ 共t兲 sជm共t兲 = uជ m共t兲 +
共s , ⬘兲
Psជ共s , ⬘ ; t兲
2
共t兲 =
冏
冏
⬁
1 兺 vជ m共t兲 = 兩具vជ 共t兲典兩, N m=1
共B4兲
APPENDIX B: ANALYTIC COMPUTATION OF THE PHASE TRANSITION FOR THE VECTORIAL NETWORK MODEL
⬁ where we have defined 具vជ 共t兲典 = N1 兺m=1 vជ m共t兲. Under the meanfield assumptions, all the vectors vជ m are equally distributed with the common probability distribution Pvជ 共v , ; t兲. Then 具vជ 共t兲典 can be computed as follows. Let Pˆvជ 共 , ␥ ; t兲 be the Fourier transform 共in polar coordinates兲 of Pvជ 共v , ; t兲 共the variables and ␥ are the polar coordinates of Fourier conjugate vector ជ corresponding to vជ 兲. A cumulant expansion of Pˆvជ 共 , ␥ ; t兲 up to the first order gives
The dynamics of the vectorial network model 共VNM兲 are given by the interaction rule
Pˆvជ 共, ␥ ;t兲 ⬇ 1 − i具vជ 共t兲典 · ជ + ¯ ,
冋兺 j=1
s
册
K
m共t + 1兲 = Angle
v
Pˆuជ 共 , ␥ ; t兲 Pˆជ共 , ␥ ; t兲
Denoting as ␣ the angle between 具vជ 共t兲典 and ជ , and using the fact that 共t兲 = 兩具vជ 共t兲典兩, Eq. 共B5兲 can be written as
vជ m j共t兲 + K1eim共t兲 + 2m共t兲,
共B1兲 兵vជ m j其Kj=1
are the K inputs of vជ m, and m共t兲 and m共t兲 are where independent random variables uniformly distributed in the interval 关− , 兴 and 0 ⱕ 1 , 2 ⱕ 1. Let us define the extrinsic noise vector nជ e and the intrinsic noise ni as nជ e = K1ei共t兲,
共B5兲
n i = 2 .
We will denote as Pnជ e共r , 兲 the PDF 共in polar coordinates兲 of the extrinsic noise nជ e and as Pni共兲 the PDF of the intrinsic noise ni. As for the majority voter model, it is convenient to define the quantities
Pˆvជ 共, ␥ ;t兲 ⬇ 1 − i共t兲 cos ␣ + ¯ .
共B6兲
Thus, a first order cumulant expansion of Pˆvជ 共 , ␥ ; t兲 directly gives us the order parameter 共t兲. The objective of the calculation is to find a recurrence relation in time for Pˆvជ 共 , ␥ ; t兲 based on Eq. 共B1兲. From this recurrence relation we will obtain the dynamical mapping that determines the temporal evolution of 共t兲. 2. Recurrence relation for Pˆvᠬ ( , ␥ ; t)
Note first that, since 兩vជ m兩 = 1 for all m, then Pvជ 共v , ; t兲 can be written as
K
uជ m共t兲 = 兺 vជ m j共t兲,
共B2兲
Pvជ 共v, ;t兲 =
j=1
sជm共t兲 = uជ m共t兲 + nជ e共t兲.
共B3兲
Let 共vm , m兲, 共um , m兲, and 共sm , m ⬘ 兲 be the polar coordinates of the vectors vជ m共t兲, uជ m共t兲, and sជm共t兲, respectively. We will denote as Pvជ m共vm , m ; t兲, Puជ m共um , m ; t兲, and Psជm共sm , m ⬘ ; t兲 the PDF’s of these three vectors, respectively. In Table I we summarize the relevant quantities appearing in this calculation. Note that neither Pnជ e共r , 兲 nor Pni共兲 depend on time or on the subscript m of vជ m, whereas the PDF’s of vជ m共t兲, uជ m共t兲, and sជm共t兲 depend on both time and the subscript m. However, as a mean-field approximation, we again assume that all the vectors vជ m statistically equivalent and statistically independent. In this case, the functions Pvជ m共vm , m ; t兲, Puជ m共um , m ; t兲, and Psជm共sm , m ⬘ ; t兲 are the same for all the vectors in the network and the subscript m can be omitted.
␦共v − 1兲 v
P共 ;t兲,
共B7兲
where P共 ; t兲 is the PDF of the angle 共t兲 of vជ 共t兲. From Eq. 共B1兲 it follows that P共 ; t兲, Psជ共s , ; t兲, and Pni共兲 are related through P共 ;t + 1兲 =
冕 冋冕
−
⬁
0
册
sPsជ共s, − ;t兲ds Pni共兲d . 共B8兲
Since sជ共t兲 = 兺Kj=1vជ m j共t兲 + nជ e共t兲, and each of the vectors vជ m j共t兲 is distributed with the PDF Pvជ 共v , ; t兲, it is clear that Psជ共s , ; t兲 depends on P共 ; t兲. Therefore, Eq. 共B8兲 is a recurrence relation in time for P共 ; t兲. This recurrent relation is best solved in Fourier space. Denoting as Pˆsជ共 , ␥ ; t兲 the Fourier transform of Psជ共s , ; t兲, the above equation can be written as
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INTRINSIC AND EXTRINSIC NOISE EFFECTS ON ...
P共 ;t + 1兲 =
1 共2兲2
冕 冕 冕 冕
⬁
d
−
⬁
d
sds
0
0
2
冋
m共t + 1兲 = 共i兲m pˆm ␦m,0m共0;t兲 + m
d␥
0
⫻Pˆsជ共, ␥ − ;t兲Pni共兲eis cos共␥−兲 .
⬁
兺 m共t兲eim , m=−⬁ m共;t兲eim共␥−兲 , 兺 m=−⬁
共B10兲
兺 pˆme−im , m=−⬁
Pˆvជ 共, ␥ ;t + 1兲 =
1 2
冕
1 m共;t兲 = 2
冕
pˆm =
1 2
共B12兲
−
−
冕
P共 ;t兲e−imd ,
Pˆsជ共, ␥ ;t兲e−im␥d␥ ,
Substituting into the above equation the form of P共 ; t兲 given in Eq. 共B10兲 共evaluated at t + 1兲, we obtain
共B19兲
共B14兲
where we have used the integral representation of the Bessel m function Jm共兲 = 共2i兲 兰20ei共mz− cos z兲dz. Now we use the value of m共t + 1兲 given in Eq. 共B17兲, which leads to
−
Pni共兲e−imd .
⬁
⬁
⫻
⬁
⬁
sds
0
dm共;t兲Jm共s兲.
0
␦m,0 +
d⬘ . ⬘
Pˆsជ共, ␥ ;t兲 = 关Pˆvជ 共, ␥ ;t兲兴KJ0共K兲.
0
␦共兲
m共⬘ ;t兲
共B20兲
共B21兲
where Pˆnជ e共 , ␥兲 is the Fourier transform of the PDF of the noise vector nជ e = K1ei. Since is uniformly distributed in the interval 关0 , 2兴, it follows that Pˆnជ e共 , ␥兲 = J0共K兲. Therefore, we obtain 共B22兲
Substituting the above expression into Eq. 共B14兲 we obtain
Exchanging the order of integration in the last expression, and using the identity sJm共s兲ds =
⬁
Pˆsជ共, ␥ ;t兲 = 关Pˆvជ 共, ␥ ;t兲兴K Pˆnជ e共, ␥兲,
共B16兲
冕
2mpˆmJm共兲eim
To complete the projection of Pˆvជ 共 , ␥ ; t兲 onto the unit circle in a closed form, it only remains to find m共t兲 as a function of Pˆvជ 共 , ␥ ; t兲. Since s共t兲 = 兺Kj=1vជ m j共t兲 + nជ e共t兲, and we are assuming that all the vជ j are statistically independent, then
0
冕 冕
冕
兺
m=−⬁
0
dm共;t兲Jm共s兲eim ,
sds
0
Pˆvជ 共, ␥ ;t + 1兲 = 2 pˆ0J0共兲 +
共B15兲
where we have used the integral representation of the Bessel m 2 i共mz+x cos z兲 function Jm共x兲 = 共−i兲 dz. It follows from the last 2 兰 0 e expression that
⬁
共− i兲mm共t + 1兲Jm共兲eim␥ , 兺 m=−⬁
⬁
冕 冕
m共t + 1兲 = im pˆm
P共 ;t + 1兲e−i cos共−␥兲d . 共B18兲
共B13兲
m共t + 1兲eim 兺 m=−⬁
兺 共i兲mpˆm m=−⬁
2
⬁
⬁
=
冕
Pˆvជ 共, ␥ ;t + 1兲 = 2
Substituting Eqs. 共B10兲 and 共B11兲 into Eq. 共B9兲, carrying out the integration over and taking into account Eq. 共B15兲 we obtain
⬁
册
0
where m共t兲, m共 ; t兲, and pˆm are given by
m共t兲 =
0
d .
Note that Eq. 共B17兲 is a consequence of the recurrence relation given in Eq. 共B8兲, which in turn follows directly from the dynamic interaction rule Eq. 共B1兲. Now we have to project the probability distribution function Psជ共s , ; t兲 onto the unit circle by forcing the vector sជ共t兲 to have unit length at time t + 1, and thus becoming vជ 共t兲. To do so, we take the Fourier transform of Pvជ 共v , ; t兲 given in Eq. 共B7兲, which when evaluated at time t + 1 gives
共B11兲
⬁
P ni共 兲 =
m共;t兲
共B17兲
⬁
Pˆsជ共, ␥ − ;t兲 =
⬁
共B9兲
Since P共 ; t兲, Pˆsជ共 , ␥ ; t兲, and Pni共兲 are periodic functions of their angular arguments 共, ␥, and respectively兲, we can expand these functions in Fourier series as P共 ;t兲 =
冕
m共;t兲 =
m , 2
1 J0共K兲 2
冕
2
关Pˆvជ 共, ␥ ;t兲兴Ke−im␥d␥ .
0
共B23兲
where ␦共兲 and ␦m,0 are the Dirac and Kronecker delta functions, respectively, Eq. 共B16兲 becomes
Finally, combining this result with Eq. 共B20兲, we obtain the desired recurrence relation for Pˆvជ 共 , ␥ ; t兲:
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PIMENTEL et al. ⬁
Pˆvជ 共, ␥ ;t + 1兲 = 2 pˆ0J0共兲 + ⫻
冕
⬁
0
⬁
兺 mpˆmJm共兲eim␥ m=−⬁
d⬘ J0共K⬘兲 ⬘
冕
2
⫻
d␥⬘ 共B24兲
3. Dynamical mapping for K \ ⴥ
Equation 共B24兲 is a complicated recurrence relation the exact solution to which is way out of our hands. However, for large values of K, namely, for a large number of interactions per particle, we can use the central limit theorem to approximate 关Pˆvជ 共 , ␥ ; t兲兴K as
再
冕
冕
⬁
0
⫻e
冕
2
冕
⫻
冕
0
冕
2
e−i关m+兩具vជ 共t兲典兩x cos 兴d
where we have used the fact that 共t兲 = 兩具vជ 共t兲典兩. Substituting this result into Eq. 共B25兲, we obtain ⬁
Pˆvជ 共, ␥ ;t + 1兲 = 2 pˆ0J0共兲 +
冕
2共− i兲mmpˆmJm共兲 兺 m=−⬁
⬁
dx J0共x兲Jm关共t兲x兴. 共B26兲 x
0
Now, recalling that Pni共兲 is a constant normalized function in the interval 关−2 , 2兴, with 0 ⱕ 2 ⱕ 1, its Fourier transform pˆm is given by .
pˆm =
sin共m2兲 . 2 2m 2
共B27兲
Thus, 2 pˆ0 = 1 and Eq. 共B26兲 can be written as ⬁
Pˆvជ 共, ␥ ;t + 1兲 = J0共兲 +
兺
共− i兲m2mpˆmJm共兲
m=−⬁
兺 mpˆmJm共兲eim␥ m=−⬁
dx J0共x兲 x
0
e−i关m␥⬘+兩具vជ 共t兲典兩x cos共␥⬘−␣兲兴d␥⬘
0
⬁
⬁
2
⫻ eim共␥−␣兲
Making the change of variable xជ = Kជ⬘ in the above expression, we obtain
Pˆvជ 共, ␥ ;t + 1兲 = 2 pˆ0J0共兲 +
d␥⬘e−i关m␥⬘+具vជ 共t兲典·xជ 兴 .
= e−im␣2共− i兲mJm关共t兲x兴,
0
e
2
= e−im␣2共− i兲mJm关兩具vជ 共t兲典兩x兴
d␥⬘
−iK具vជ 共t兲典·ជ⬘−共K/2兲ជ⬘·C共t兲·ជ⬘ −im␥⬘
e−i关m␥⬘+具vជ 共t兲典·xជ 兴d␥⬘ =
= e−im␣
mpˆmJm共兲eim␥
d⬘ J0共K⬘兲 ⬘
2
0
⬁
⫻
冕
0
where 具vជ 共t兲典 and C共t兲 are the first moment and covariance matrix of Pvជ 共v , ; t兲, respectively. With this approximation, Eq. 共B24兲 becomes
兺
dx J0共x兲 x
Now, we can write 具vជ 共t兲典 · xជ = 兩具vជ 共t兲典兩x cos共␥⬘ − ␣兲, where ␥⬘ and ␣ are the angles in Fourier space of xជ and 具vជ 共t兲典, respectively. The second integral on the right-hand side of Eq. 共B25兲 becomes
冎
m=−⬁
⬁
共B25兲
K 关Pˆvជ 共, ␥ ;t兲兴K ⬇ exp − iK具vជ 共t兲典 · ជ − ជ · C共t兲 · ជ , 2
Pˆvជ 共, ␥ ;t + 1兲 = 2 pˆ0J0共兲 +
冕
0
0
⫻关Pˆvជ 共⬘, ␥⬘ ;t兲兴Ke−im␥⬘ .
兺 mpˆmJm共兲eim␥ m=−⬁
Pˆvជ 共, ␥ ;t + 1兲 = 2 pˆ0J0共兲 +
冕
2
⫻ eim共␥−␣兲
冕
⬁
0
dx J0共x兲Jm关共t兲x兴. x 共B28兲
d␥⬘
Finally, expanding both sides of the above equation up to the first order in , and recalling Eq. 共B6兲 for the left-hand side, we obtain the recurrence relation for the order parameter
0
⫻e−i具vជ 共t兲典·xជ −xជ ·C共t兲·xជ /2Ke−im␥⬘ . We can go a step further in the large-K approximation and 1 xជ · C共t兲 · xជ appearing in the exponent inside neglect the term 2K the integral of the last expression, which gives
共t + 1兲 = 2 pˆ1
冕
⬁
J0共x兲J1关共t兲x兴
0
This is Eq. 共22兲 of the main text.
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共B29兲
PHYSICAL REVIEW E 77, 061138 共2008兲
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061138-17
−⬁
⬁ ˆ = 共2兲−1兰−⬁ f 共兲e−ixd for the Fourier transform and its inverse.
